PHYS-333: Fundamentals of Astrophysics

Stan Owocki

Department of Physics & Astronomy, University of Delaware, Newark, DE 19716

Version of March 19, 2018

II. STELLAR STRUCTURE & EVOLUTION

Contents

13 Balance between Pressure and Gravity 13.0

13.1 Hydrostatic equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.0

13.2 Pressure scale height and thinness of surface layer . . . . . . . . . . . . . . . 13.2

13.3 Hydrostatic balance in stellar interior and the virial temperature . . . . . . . 13.3

13.4 Questions and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4

14 Transport of Radiation from Interior to Surface 14.1

14.1 Random walk of photon diffusion from stellar core to surface . . . . . . . . . 14.1

14.2 Diffusion approximation at depth . . . . . . . . . . . . . . . . . . . . . . . . 14.3

14.3 Atmospheric variation of temperature with optical depth . . . . . . . . . . . 14.4

14.4 Questions and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4

15 Structure of Radiative vs. Convective Stellar Envelopes 15.1

15.1 L ∼M3 relation for hydrostatic, radiative stellar envelopes . . . . . . . . . . 15.1

15.2 Horizontal-track Kelvin-Helmholtz contraction to the main sequence . . . . . 15.2

15.3 Convective instability and energy transport . . . . . . . . . . . . . . . . . . . 15.2

15.4 Fully convective stars – the Hayashi track for proto-stellar contraction . . . . 15.5

– 0.2 –

16 Hydrogen Fusion and the Mass Range of Stars 16.1

16.1 Core temperature for H-fusion . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2

16.2 Main sequence scalings for radius-mass and luminosity-temperature . . . . . 16.3

16.3 Lower mass limit for hydrogen fusion: Brown Dwarf stars . . . . . . . . . . . 16.4

16.4 Upper mass limit for stars: the Eddington Limit . . . . . . . . . . . . . . . . 16.5

17 Post-Main-Sequence Evolution 17.1

17.1 Post-MS evolution and death of solar-type stars with M . 8M . . . . . . . 17.1

17.1.1 Core-Hydrogen burning and evolution to the Red Giant branch . . . 17.1

17.1.2 Helium flash and core-Helium burning on the Horizontal Branch . . . 17.2

17.1.3 Asymptotic Giant Branch to Planetary Nebula to White Dwarf . . . 17.4

17.1.4 White Dwarf stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.5

17.1.5 Chandasekhar limit for white-dwarf mass: M < 1.4M . . . . . . . . 17.6

17.2 Post-MS evolution and death of high-mass stars with M > 8M . . . . . . . 17.7

17.2.1 Multiple shell burning and horizontal loops in H-R diagram . . . . . 17.7

17.2.2 Core-collapse supernovae . . . . . . . . . . . . . . . . . . . . . . . . . 17.8

17.2.3 Neutron stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.10

17.2.4 Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.10

17.3 Observations of stellar remnants . . . . . . . . . . . . . . . . . . . . . . . . . 17.13

C Radiative Transfer C.1

C.1 Absorption and thermal emission in a stellar atmosphere . . . . . . . . . . . C.1

C.2 The Eddington-Barbier relation for emergent intensity . . . . . . . . . . . . C.3

C.3 Questions and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.4

D Atomic origins of opacity D.0

D.1 Thomson cross-section and opacity for free electron scattering . . . . . . . . D.0

– 0.3 –

D.2 Atomic absorption and emission: free-free, bound-bound, bound-free . . . . . D.2

D.3 Questions and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.3

– 13.0 –

Part II: Stellar Structure and Evolution

We have seen in part I how a star’s color or peak wavelength λmax indicates its charac-

teristic temperature near the stellar surface. But what about the temperature in the star’s

deep interior? Intuitively, we expect this to be much higher than at the surface, but under

what conditions does it become hot enough to allow for nuclear fusion to power the star’s

luminosity? And how does it scale quantitatively with the overall stellar properties, like

mass M , radius R, and perhaps luminosity L?

To answer these questions, let us identify two distinct considerations for our intuition

that the interior temperature should be much higher than at the surface.

The first we might characterize as the “blanketing” by the overlying layers, which traps

any energy generated in the interior, much as a blanket in bed traps our body heat, keeping

our skin temperature at a comfortable warmth, instead of the relative chilliness of having it

exposed to open air. In this picture, the equilibrium interior temperature depends on the

rate of energy generation (from metabolism for our bodies, or nuclear fusion for stars) and

the ‘insulation thickness’ of the overlying of material to the surface (given by the optical

depth; see §14.)

But distinct from this consideration of the transport of energy from the interior, there is

for a star a dynamical requirement for force or momentum balance, to keep the star supported

against the inward pull of its own self-gravity. Since stars are gaseous, without the tensile

strength of a solid body, this gravitational support is supplied by increased internal gas

pressure P , allowing the star to remain in a static equilibrium. This high gas pressure arises

from a combination of high density and high temperature. As detailed in §13.3, this allows

us to determine a characteristic interior temperature, through a further application of the

Virial theorem for bound systems that was briefly discussed for bound orbits in part I.

13. Balance between Pressure and Gravity

13.1. Hydrostatic equilibrium

To quantify this gravitational equilibrium for a static star, consider, as illustrated in

figure 13.1, a thin radial segment of thickness dr with local density ρ and downward gravita-

tional acceleration g. The mass-per-unit-area of this layer is dm = ρdr, with corresponding

weight-per-unit-area g dm. To support this weight, the gas pressure at the lower end of this

– 13.1 –

P(r+dr)r+dr

P(r)rdP = −ρ gdr

dP__ −ρg dr

Fig. 13.1.— Illustration of the radial decline of gas pressure P due to local mass density ρ

and downward gravity g.

layer must be higher by amount |dP | = g dm = ρg dr than the upper end, implying

dP

dr= −ρ g , (13.1)

a condition know as Hydrostatic Equilibrium.

For an ideal gas, the pressure depends on the product of the number density n = ρ/µ

and temperature T ,

P = nkT = ρkT

µ≡ ρc2

s , (13.2)

where k = 1.38 × 10−16 erg/K is Boltzmann’s constant, µ is the average mass – a.k.a. the

“mean molecular weight” – of all particles (i.e., both ions and electrons) in the gas, and the

final equation defines the isothermal1 sound speed, cs ≡√kT/µ .

For any given element, the fully ionized molecular weight is just set by the nuclear mass

Amp (because the electron mass is by comparison negligible) divided by the nuclear charge

number plus one (for the one nucleus + Zn electrons that balance the nuclear charge eZn),

µ = mpA/(Zn+1). For a gas mixture with mass fraction X, Y , and Z for H, He, and metals,

1This speed, which was first derived by Newton, would only be the speed of sound if the gas remained

strictly constant temperature (isothermal). In practice, the temperature fluctuations associated with the

gas compressions make the actual “adiabatic” speed of sound slightly higher, by a factor√γ, where γ is the

ratio of specific heats (5/3 for a monatomic gas).

– 13.2 –

the overall mean molecular weight is then obtained by a weighted average of the inverses

(mp/µ = (Zn + 1)/A) of the individual components (as in a parallel circuit), yielding

µ =mp

2X + 3Y/4 + Z/2≈ 0.6mp ≡ µ , (13.3)

where the last equality is for the solar case with X = 0.72, Y = 0.26, and Z = 0.02. More

generally, for fully ionized gases the proton-mass-scaled molecular weight µ/mp can range

from 1/2 for pure H (X = 1), to 4/3 for pure He (Y = 1), to a maximum of 2 for pure heavy

metals (Z = 1).

13.2. Pressure scale height and thinness of surface layer

The ratio of eqns. (13.2) to (13.1) defines a characteristic pressure scale height,

H ≡ P

|dP/dr|=kT

µg=c2

s

g, (13.4)

where the absolute value of the pressure gradient dP/dr (which itself is negative) ensures

the scale height is positive.

At the stellar surface radius r = R, where the gravity and temperature approach their

fixed surface values g∗ = GM/R2 and T = T∗, the scale height becomes quite small, typically

only a tiny fraction of the stellar radius,

H

R=

kT∗/µ

GM/R=

2c2s∗

V 2esc

≈ 0.0005T∗/Tµ/µ

R/RM/M

, (13.5)

where Vesc is the escape speed introduced in part I. For the solar atmosphere, the sound

speed is cs∗ ≈ 9 km/s, about 1/60th of the surface escape speed Vesc = 620 km/s.

If we further idealize a stellar atmosphere as being roughly isothermal, i.e. with a nearly

constant temperature T ≈ T∗, then, since the gravity is also effectively fixed at the surface

value, we see that the scale height also becomes constant. This makes it easy to integrate

the hydrostatic equilibrium equation (13.1), thus giving the variation of density and pressure

in terms of a simple exponential stratification with height z ≡ r −R ,

P (z)

P∗=ρ(z)

ρ∗= e−z/H , (13.6)

where the asterisk subscripts denote values at some some surface layer where z ≡ 0 (or r =

R). In practice the temperature variations in an atmosphere are gradual enough that quite

generally both pressure and density very nearly follow such an exponential stratification.

– 13.3 –

The results in this subsection actually apply to any gravitationally bound atmosphere,

not only for stars but also for planets, including the earth, with similarly small characteristic

values for the ratio H/R. This is the basic reason that the earth’s atmosphere is confined to

such a narrow layer around its solid surface, meaning that at just a couple hundred kilometers

altitude it is nearly a vacuum, so tenuous that it is imparts only a weak drag on orbiting

satellites.

For stars or gaseous giant planets without a solid surface, it means that the dense, opaque

regions have only a similarly narrow transition to the fully transparent upper layers, thus

giving them a similarly sharp visual edge as a solid body. For stars it means that models

of the escape of interior radiation through this narrow atmospheric layer can essentially

ignore the stellar radius, allowing the emergent spectrum to be well described by a planar

atmospheric model fixed by just two parameters – surface temperature and gravity– and not

dependent on the actual stellar radius.

13.3. Hydrostatic balance in stellar interior and the virial temperature

This hydrostatic balance must also apply in the stellar interior, but now both the tem-

perature and gravity have a strong spatial variation. At any given interior radius r, the local

gravitational acceleration depends only on the mass within that radius,

M(r) ≡ 4π

∫ r

0

ρ(r′)r′2 dr′ . (13.7)

This thus requires the hydrostatic equilibrium equation to be written in the somewhat more

general form,

dP

dr= −ρ(r)

GM(r)

r2. (13.8)

This represents one of the key equations for stellar structure.

The implications of hydrostatic equilibrium for the hot interior of stars are quite different

from the steep exponential pressure drop near the surface; indeed they allow us now to derive

a remarkably simple scaling relation for a characteristic interior temperature Tint.

For this consider the associated interior pressure Pint at the center of the star (r =

0); to drop from this high central pressure to the near-zero pressure at the surface, the

pressure gradient averaged over the whole star must be |dP/dr| ≈ Pint/R. We can similarly

characterize the gravitational attraction in terms of the surface gravity g∗ = GM/R2 times

an interior density that scales as ρint ∼ Pintµ/kTint. Applying these in the basic definition

– 13.4 –

of scale height (13.4), we find that for the interior H ≈ R, which in turn implies for this

characteristic stellar interior temperature,

Tint ≈GMµ

kR≈ 13× 106K

M/MR/R

. (13.9)

Thus, while surface temperatures of stars are typically a few thousand Kelvin, we see that

their interior temperatures are typically of order 10 million Kelvin! As discussed below (§16),

this is indeed near the temperature needed for nuclear fusion of Hydrogen into Helium in

the stellar core.

This close connection between thermal energy of the interior (∼ kT ) to the star’s grav-

itational binding energy (∼ GMµ/R) is really just another example of the Virial theorum

for gravitationally bound systems, as discussed in part I for the case of bound orbits. The

temperature is effectively a measure of the average kinetic energy associated with the ran-

dom thermal motion of the particles in the gas. Thermal energy is thus just a specific form

of kinetic energy, and the Virial theorum tells us that the average kinetic energy in a bound

system equals one-half the magnitude of the gravitational binding energy.

13.4. Questions and Exercises

Quick Question 1:

(a.) For a typical temperature on a spring day (∼ 50oF), compute the scale

height H for the earth (in km), and its ratio to earth’s radius, H/Re.

(b.) Relative to values at sea level, compute the pressure and density at a typical

height h = 300 km for an orbiting satellite.

Quick Question 2:

(a.) Compute the escape speed (in km/s) from stars with M = M and R = 2R;

with M = 2M and R = R.

(b.) For these stars, estimate the associated central temperature.

– 14.1 –

14. Transport of Radiation from Interior to Surface

14.1. Random walk of photon diffusion from stellar core to surface

Let us next turn to the “blanketing” effect of the star’s material in trapping the heat

and radiation of the interior. Within a star the absorption of light by stellar material is

counteracted by thermal emission. As illustrated in figure 14.1, radiation generated in the

deep interior of a star undergoes a diffusion between multiple encounters with the stellar

material before it can escape freely into space from the stellar surface. The number of mean-

free-paths ` from the center at r = 0 to the surface at radius r = R now defines the central

optical depth

τc =

∫ R

0

dr′

`=

∫ R

0

κρ dr′ . (14.1)

As discussed in Appendix D, the opacity in stellar interiors typically has a CGS value

of κ ≈ 1 cm2/g, with a minimum set by the value for Thomson scattering by free electrons,

κe ≈ 0.34 cm2/g. We can then estimate a typical value of this central optical depth by simply

taking the density to be roughly characterized by its volume average ρ = M/(4πR3/3),

where again M and R are the stellar mass and radius. For the Sun this works out to give

ρ ≈ 1.4 g/cm3, i.e. just above the density of water. (The Sun wouldn’t quite float in your

bathtub.) Since the opacity is also near unity in CGS units, the average mean-free-path for

scattering in the Sun is just ¯ ≈ 0.7 cm.

In the core of the actual Sun, the density is typically a hundred times higher than this

mean value, so the core mean-free-path is a factor hundred smaller, i.e `core ≈ 0.07 mm! But

either way, the mean-free-path is much, much smaller than the solar radius R ≈ 700, 000 km

= 7× 1010 cm. This implies the optical depth from the center to surface is truly enormous,

with a typical value

τc ≈R¯≈ 1011 . (14.2)

The total number of scatterings needed to diffuse from the center to the surface can

then be estimated from a basic “random walk” argument. The simple 1D version states that

after N left/right random steps of unit length, the root-mean-square (rms) distance from

the origin is√N . For the 3D case of stellar diffusion, this rms number of unit steps can be

roughly associated with the total number of mean-free-paths between the core and surface,

i.e.√N ≈ τ . This implies that photons created in the core of the Sun need to scatter a total

of N ≈ τ 2 ≈ 1022 times to reach the surface!

In traveling from the Sun’s center to its surface, the net distance is just the Sun’s radius

R; but the cumulative path length traveled is much longer, `tot ≈ N ¯ ≈ τ 2 ¯ ≈ τR. For

– 14.2 –

Fig. 14.1.— Illustration of the random-walk diffusion of photons from the core to surface

of a star.

photons traveling at the speed of light c = 3×1010cm/s, the total time for photons to diffuse

from the center to the surface is thus

tdiff = τ 2¯c≈ τ

Rc≈ 1011 × 2.3 s ≈ 7000 yr , (14.3)

where for the last evaluation, it is handy to recall again that 1 yr≈ π × 107 s.

Once the photons reach the surface, they can escape the star and travel unimpeded

through space, taking, for example, only a modest time tearth = au/c≈ 8 min to cross the

1 au (≈ 215R) distance from the sun to the earth.

A stellar atmospheric surface thus marks a quite distinct boundary between the interior

and free space. From deep within the interior, the stellar radiation field would appear nearly

isotropic (same in all directions), with only a small asymmetry (of order 1/τ) between upward

and downward photons. But near the surface, this radiation becomes distinctly anisotropic,

emerging upward from the surface below, but with no radiation coming downward from

empty space above.

– 14.3 –

14.2. Diffusion approximation at depth

This picture of photons undergoing a random walk through the stellar interior can be

formalized in terms of a diffusion model for radiation transport in the interior. Appendix

C discusses the transition from diffusion to free-streaming that occurs in the narrow region

near the stellar surface, known as the “stellar atmosphere”. This is described by the equation

of radiative transfer, given by eqn. (C1), with eqn. (C2) now defining the vertical optical

depth τ(r) from a given radius r to an external observer at r →∞.

But in the deep interior layers within a star, i.e. with large optical depths τ 1, the

trapping of the radiation makes the intensity I nearly isotropic and near the local Planck

function B. Applying this to the derivative term in eqn. (C1) and solving for I gives an

“diffusion approximation” form for the intensity,

I(µ, τ) ≈ B(τ) + µdB

dτ, (14.4)

where we recall from §12.1 that µ is the cosine of the angle between the ray and the vertical

direction, so that µ = +1 is directly upward, and µ = −1 is directly downward.

Since dB/dτ is of order B/τ , we can see that the second term is much smaller, by a factor

∼ 1/τ 1, than the first, leading-order term. Recall that both the specific intensity I and

the Planck functionB have the same units as a surface brightness, i.e. energy/area/time/solid

angle.

The local net upward flux F (energy/area/time) is computed by weighting the intensity

by the direction cosine µ and then integrating over solid angle,

F ≡∮Iµ dΩ = 2π

∫ +1

−1

(µB(τ) + µ2dB

dτ

)dµ . (14.5)

Since the leading order term with B(τ) is then odd over the range −1 < µ < 1, it vanishes

upon integration, giving

F ≈ 4π

3

dB

dτ. (14.6)

Again recalling that B = σsbT4/π, and noting the optical depth changes with radius r as

dτ = −κρdr, we can alternatively write the flux as a function of the local temperature

gradient,

F (r) = −[

4π

3κρ

∂B

∂T

]dT

dr= −

[16σsb3κρ

T 3

]dT

dr. (14.7)

The terms in square bracket can be thought of as a radiative conductivity, which we note

increases with the cube of the temperature T 3, but depends inversely on opacity and density,

1/κρ.

– 14.4 –

14.3. Atmospheric variation of temperature with optical depth

A star’s luminosity L is generated in a very hot, dense central core. Outside this core,

at any stellar envelope radius r, the local radiative flux scales as F = L/4πr2, which near

the stellar surface r . R approaches the fixed surface value F∗ = L/4πR2 ≡ σsbT4eff , where

the last equation recalls the definition of the stellar effective temperature Teff . Since in such

a surface layer F∗ is independent of τ , eqn. (14.6) can be trivially integrated in this layer to

give,4π

3B(τ) = F∗τ + C = σsbT

4effτ + C , (14.8)

where C is an integration constant. Recalling also from eqn. (5.1) that πB = σsbT4, we can

convert (14.8) into an explicit expression for the variation of temperature with optical depth,

T 4(τ) =3

4T 4

eff [τ + 2/3] , (14.9)

wherein, in light of the result in §C.2, we have taken the integration constant such that

T (τ = 2/3) = Teff .

Together with the equation (13.1) for hydrostatic equilibrium, equation (14.7) for radia-

tive diffusion determines the fundamental structure of the stellar interior. Let us next apply

these to explain the empirical mass-luminosity relation L ∼M3.

14.4. Questions and Exercises

Quick Question 1:

a. At what optical depth τ does the local temperature T in a stellar atmosphere

equal the stellar effective temperature Teff?

b. At about what optical depth τ does the local temperature T = 10Teff?

Quick Question 2:

a. Near the Sun’s surface where the temperature is at the effective temperature

T = T∗ = Teff ≈ 5800 K, compute the scale height H (in km).

b. Using the fact that the mean-free-path ` ≈ H near this surface, compute the

mass density ρ (in g/cm3) assuming the opacity is equal to the electron scattering

value given in §D.1, i.e. κe = 0.34 cm2/g.

– 15.1 –

15. Structure of Radiative vs. Convective Stellar Envelopes

15.1. L ∼M3 relation for hydrostatic, radiative stellar envelopes

As discussed in part I, observations of binary systems show that main sequence stars

follow an empirical mass-luminosity relation L ∼ M3. The physical basis for this can be

understood by considering the two basic relations of stellar structure, namely hydrostatic

equilibrium and radiative diffusion, as given in eqns. (13.1) and (14.7) above.

As in the Virial scaling for internal temperature given in §13.3, we can use a single

point evaluation of the hydrostatic pressure gradient to derive a scaling between interior

temperature T , stellar radius R and mass M , and molecular weight µ,

dP

dr= −ρGMr

r2

ρT

µR∼ ρ

M

R2

T R ∼ M µ , (15.1)

Likewise, a single point evaluation of the temperature gradient in the radiative diffusion

equation (14.7) gives

F = −16σsb3κρ

T 3 dT

dr

L

R2∼ R3

κM

T 4

R

L ∼ (RT )4

κM

L ∼ M3 µ4

κ, (15.2)

where the last scaling uses the hydrostatic equilibrium scaling in (15.1) to derive the basic

scaling law L ∼M3, assuming a fixed molecular weight µ and stellar opacity κ.

Two remarkable aspects of this derivation are that (1) the role of the stellar radius

cancels; and (2) the resulting M − L scaling does not depend on the details of the nuclear

generation of the luminosity in the stellar core! Indeed, this scaling was understood from

stellar structure analyses that were done (e.g. by Eddington, and Schwarzschild) in the

– 15.2 –

1920’s, long before Hydrogen fusion was firmly established as a key energy source for the

sun and other main-sequence stars (e.g., by Hans Bethe ca. 1939).

15.2. Horizontal-track Kelvin-Helmholtz contraction to the main sequence

In fact, this simple L ∼M3 scaling even applies to the final stages of pre-main-sequence

evolution, when the core is not yet hot enough to start nuclear burning, but the envelope

has become hot enough for radiative diffusion to dominate the transport of energy generated

by the star’s gravitational contraction. As the radius decreases over the Kelvin-Helmholtz

timescale tKH of this contraction, the surface temperature increases in a way that keeps

the luminosity nearly constant. The lower panel of figure 15.1 illustrates that, on the H-R

diagram, a late-phase pre-main-sequence star thus evolves along a horizontal track from right

to left, stopping when it reaches the main sequence; this is where the core temperature is

now high enough for H-fusion to take over in supplying the energy for the stellar luminosity,

without any need for further contraction. As discussed in §16, for a given mass, a star’s

radius on the main sequence is just the value for which the interior temperature, as set by

the Virial theorem, is sufficiently high to allow this H-fusion in the core.

15.3. Convective instability and energy transport

In practice, the transport of energy from the stellar interior toward the surface sometimes

occurs through convection instead of radiative diffusion, and this has important consequence

for stellar structure and thus for the scaling of luminosity.

Convection refers to the overturning motions of the gas, much like the bubbling of

boiling water on a stove. Stars become unstable to forming convection whenever the processes

controlling the temperature make its spatial gradient too steep. This can occur in the nuclear

burning core of massive stars, for which the specific mechanism for Hydrogen fusion, called

the “CNO” cycle, gives the nuclear burning rate a steep dependence on temperature. The

resulting steep temperature gradient makes the cores of such stars strongly convective.

Steep gradients, and their associated convection, can also occur in outer regions of

cooler, lower-mass stars, where the cooler temperature induces recombination of ionized H

or He. The bound-free absorption by this neutral Hydrogen significantly increases the local

stellar opacity κ. For a fixed stellar flux F = L/4πr2 of stellar luminosity L that needs to

be transported through an interior radius r, the radiative diffusion eqn. (14.7) shows that

– 15.3 –

Fig. 15.1.— Illustration of the pre-main-sequence evolution of stars. The upper panel

shows how during the early stages of a collapsing proto-star, the interior is fully convective,

causing it to evolve with decreasing luminosity at a nearly constant, relatively cool surface

temperature, and so down the nearly vertical “Hayashi track” in the H-R diagram. The

lower panel shows the final approach to the main sequence for stars of various masses. For

stars with a solar mass or above, the stellar interior becomes radiative, stopping the Hayashi

track decline in luminosity. The stars then evolve horizontally and to the left on the H-

R diagram, each with fixed luminosity but increasing temperature, till they reach their

respective positions on the “zero-age-main-sequence” or ZAMS, when the core is hot enough

to ignite H-fusion.

– 15.4 –

the required radiative temperature gradient increases with such increased opacity,∣∣∣∣dTdr∣∣∣∣rad

=3κρF

16σsbT 3∼ κ (15.3)

r0

r1=r0+δrρ1,T1

P’(r)= P(r)

ρ1’,T1’

ρ0’,T0’ = ρ0,T0

ρ’T’= ρT

Fig. 15.2.— Illustration of upward displacement of a spherical fluid element in test for

convective instability, which occurs when the displaced element has a lower density ρ′1 than

that of its surroundings, ρ1. Since the pressure must remain equal inside and outside the

element, this requires the element to have a higher temperature, T ′1 > T1. Since the overall

temperature gradient is negative, convection thus occurs whenever the magnitude of the

atmospheric temperature gradient is steeper than the adiabatic gradient that applies for the

adiabatically displaced element, i.e., |dT/dr| > |dT/dr|ad.

If this gradient becomes too steep, then, as illustrated in figure 15.2, a small element

of gas that is displaced slightly upward becomes less dense than its surroundings, giving it

a buoyancy that causes it to rise higher still. A key assumption is that this dynamical rise

of the fluid occurs much more rapidly than the rate for energy to diffuse into or out of the

gas element. Processes that occur without any such energy exchange with the surroundings

are called “adiabatic”, with a fixed (power-law) relation of pressure with density or temper-

ature. In a hydrostatic medium with a set pressure gradient, this implies a fixed adiabatic

temperature gradient (dT/dr)ad.

– 15.5 –

Starting from an initial radius r0 with equal density and temperature inside and outside

some chosen fluid element (i.e., ρ′0 = ρ, T ′0 = T0), let us determine the density ρ′1 of that

element after it is adiabatically displaced to a slightly higher radius r1 = r0 + δr, where the

ambient density is ρ1. Since dynamical balance requires the element and its surrounding to

still have equal pressure after the displacement (i.e., P ′1 = P1), we have by the perfect gas law

that ρ′1T′1 = ρ1T1. If this upward displacement δr > 0 makes the element buoyant, with lower

density ρ′1 than that of it’s surroundings ρ1, then using this constant pressure condition, we

can derive the condition for the temperature gradient required for the associated convective

instability,

T1

T ′1=

ρ′1ρ1

< 1 ; Convective instability

T0 + ∆r(dT/dr)rad = T1 < T ′1 = T0 + ∆r(dT/dr)ad∣∣∣∣dTdr∣∣∣∣rad

>

∣∣∣∣dTdr∣∣∣∣ad

, (15.4)

where since both temperature gradients are negative, the condition in terms of absolute value

requires a reversal of the inequality.

We thus see that convection will ensue whenever the magnitude of the radiative tem-

perature gradient exceeds that of the adiabatic temperature gradient.

Convection is an inherently complex, 3D dynamical process that generally requires elab-

orate computer simulations to model accurately. A heuristic, semi-analytic model called

“mixing length theory” has been extensively developed, but it has serious limitations, espe-

cially near the stellar surface, where the lower density and temperature can make convective

transport quite inefficient. By contrast, in the dense and hot stellar interior, once convection

sets in, it is so efficient at transporting energy that it keeps the local temperature gradient

very close to the adiabatic value above which it is triggered.

One can thus just presume that the temperature gradient in interior convection regions

is at the adiabatic value.

15.4. Fully convective stars – the Hayashi track for proto-stellar contraction

In hot stars with T > 10, 000 K, Hydrogen remains fully ionized even to the surface;

since there then is no recombination zone to increase the opacity and trigger convection, the

energy transport in their stellar envelopes is by radiative diffusion. In moderately cooler stars

– 15.6 –

like the sun (with T ≈ 6000 K), Hydrogen recombination in a zone just somewhat below

the surface induces convection, which thus provides the final transport of energy toward the

surface; but since the deeper interior remains ionized and thus non-convective, the general

scaling laws derived assuming radiative transport still roughly apply for such solar-type stars.

However, in much cooler stars, with surface temperatures T ≈ 3500 − 4000 K, the

Hydrogen recombination extends deeper into the interior; this and other factors keep the

opacity high enough to make the entire star convectively unstable right down to the stellar

core. Because convection is so much more efficient than radiative diffusion, it can readily

bring to the surface any energy generated in the interior – whether produced by gravitational

contraction of the envelope, or by nuclear fusion in the core. As such, fully convective stars

can have luminosities that greatly exceed the value implied by the L ∼ M3 scaling law

derived in §15.1 (see eqn. 15.2) for stars with radiative envelopes. As discussed in §17, this

is a key factor in the high luminosity of cool giant stars that form in the post-main-sequence

phases after the exhaustion of Hydrogen fuel in the core.

But it also helps explain the high luminosity of the very cool, early stage of pre-main-

sequence evolution, when gravitational contraction of a large proto-stellar cloud is providing

the energy to make the cloud shine as a proto-star. Once the internal pressure generated is

sufficient to establish hydro-static equilibrium, its interior becomes fully convective, forcing

the proto-star to have this characteristic surface temperature around T ≈ 3500− 4000 K.

At early stages the proto-star’s radius is very large, meaning it has a very large lu-

minosity L = σsbT44πR2. As it contracts, it stays at this temperature, but the declining

radius means a declining luminosity. As illustrated in figure 15.1, during this early phase of

gravitational contraction, the proto-star thus evolves down a nearly vertical line in the H-R

diagram, dubbed the “Hayashi” track, after the Japanese scientist who first discovered its

significance.

Once the radius reaches a level at which the luminosity is near the value predicted by the

L ∼M3 law, the interior switches from convective to radiative, and so the final contraction to

the main sequence makes a sharp turn to a horizontal track (sometimes called the “Henyey”

track) with nearly constant luminosity but decreasing surface temperature. The luminosity

of this track is set by the stellar mass, according to the L ∼ M3 law derived for stars with

interior energy transport by radiative diffusion. The contraction is halted when the core

reaches a temperature (derived in the next section) for H-fusion, which then stably supplies

the luminosity for the main-sequence lifetime.

As detailed in §17, once the star runs out of Hydrogen fuel in its core, its post-main-

sequence evolution effectively traces backwards along nearly the same track followed during

– 15.7 –

this pre-main-sequence, ultimately leading to the cool, red giant stars seen in the upper right

of the H-R diagram.

– 16.1 –

16. Hydrogen Fusion and the Mass Range of Stars

The timescale analyses in part I (§10) show that nuclear fusion of Hydrogen into Helium

provides a long-lasting energy source that we can associate with main sequence stars in the

H-R diagram (§6.3). But what are the requirements for such fusion to occur in the stellar

core? And how is this to be related to the luminosity vs. surface temperature scaling for

main sequence stars in the HR diagram? In particular, how might this determine the relation

between mass and radius? Finally, what does it imply about the lower mass limit for stars

to undergo Hydrogen fusion?

Fig. 16.1.— The dominant reaction channel for the proton-proton chain that characterizes

hydrogen fusion in the sun and other low-mass stars.

– 16.2 –

16.1. Core temperature for H-fusion

In stars of a few solar masses or less,2 Hydrogen fusion occurs through direct proton-

proton collision, known thus as p-p ‘burning’. Figure 16.1 illustrates the most important of

the detailed reaction channels, but the overall result is simply

4 1H+ → 4He

+2 + ν + 2e+ + Eγ , (16.1)

where ν represents a weakly interacting neutrino (which simply escapes the star). The 2e+

represents two positively charged “anti-electrons”, or positrons, which quickly annihilate with

ordinary electrons, releasing ∼ 2×2× 12≈ 2 MeV of energy. The rest of the net ∼ 4×7MeV

in energy, representing the mass-energy difference between 4H vs. one He, is released as

high-energy photons (γ-rays) of energy Eγ.

The essential requirement for such p-p fusion is that the thermal kinetic energy kT of

the protons overcome the mutual repulsion of their positive charge +e, to bring the protons

to a close separation at which the strong nuclear (attractive) force is able take over, and bind

the protons together. For a given temperature T , the minimum separation b for two protons

colliding head-on comes from setting this thermal kinetic energy equal to the electrostatic

repulsion energy,

kT =e2

b. (16.2)

In particular, if we were to require that this minimum separation be equal to the size of a

Helium nucleus, i.e. b ≈ 1 fm = 10−15 m, then from eqn. (16.2) we find that the required

temperature is quite extreme, T ≈ 1.7× 1010 K!

Comparison with the virial scaling (13.9) shows this is more than a thousand times the

characteristic virial temperature for the solar interior, Tint ≈ 13 MK. As such, the closest

distance b between protons in the interior core of the sun is actually more than a thousand

times the size of the Helium nucleus, which is thus well outside the scale for operation of the

strong nuclear force that keeps the nucleus bound.

The reason that nuclear fusion can nonetheless proceed at such a relatively modest

temperature stems again from the uncertainty principle of modern quantum physics. Namely,

a proton with thermal energy mpv2th/2 = kT has an associated momentum p = mpvth =√

2mpkT . Within quantum mechanics, it thus has an associated ‘fuzziness’ in position,

characterized by its De Broglie wavelength λ ≡ h/p, where h is Planck’s constant. If λ ∼> b,

2In higher mass stars higher core temperatures, H-fusion is catalyzed by nuclear reactions of Hydrogen

with Carbon, Nitrogen, and Oxygen through what is known as the CNO cycle.

– 16.3 –

then there is a good probability that this waviness of protons will allow them to ‘tunnel’

through the electrostatic repulsion barrier between them, and so find themselves within

a nuclear distance at which the strong attractive nuclear force can bind them. Setting

b = λ = h/(mpvth) in eqn. (16.2), we can thus obtain an explicit expression for the proton

thermal speed needed for nuclear fusion of Hydrogen3,

vth,nuc =2e2

h= 690 km/s . (16.3)

Two remarkable aspects of eqn. (16.3) are: (1) this thermal speed for H-fusion depends

only on the fundamental physics constants e and h, and (2) its numerical value is very

nearly equal to the surface escape speed from the sun, vesc =√

2GM/R = 618 km/s.

Recalling the virial scaling (13.9) that says the thermal energy in the stellar interior is

comparable to the gravitational binding energy, this means that given the solar mass Mthe sun has adjusted to just the radius needed for the gravitational binding to give an interior

temperature that is hot enough for Hydrogen fusion. For mean molecular weight µ ≈ 0.6mp,

the mean thermal speed (16.3) implies a core temperature

Tnuc =µv2

th,nuc

2k= 1.2

mpe4

kh2≈ 17 MK , (16.4)

which now is quite comparable to the interior temperature Tint,vir ≈ 13 MK obtained by

applying the virial scaling (13.9) to the sun.

16.2. Main sequence scalings for radius-mass and luminosity-temperature

If we were to naively apply these same scalings to stars with different masses, then it

would suggest all stars along the main sequence should have the same, solar ratio of mass

to radius, and thus that the radius should increase linearly with mass, R ∼M .

In practice, the radius-mass relation for main-sequence stars is somewhat sublinear,

R ∼M0.7 . (16.5)

This can be understood by considering that the much higher luminosity of more massive

stars, scaling as L ∼ M3, means that the core – within which the total fuel available scales

3I am indebted to Prof. D. Mullan for pointing out to me this remarkably simple scaling

– 16.4 –

just linearly with stellar mass M – must have more vigorous nuclear burning4. The higher

core temperature to drive such more vigorous H-fusion then requires by the Virial theorum

that the mass to radius ratio of such stars must be somewhat higher than for lower mass

stars like the sun.

Combining such a sublinear radius-mass scaling R ∼ M0.7 with the mass-luminosity

scaling L ∼ M3 (eqn. 15.2) and the Stefan-Boltzmann relation L ∼ T 4R2, we infer that

luminosity should be a quite steep function of surface temperature along the main sequence,

viz. L ∼ T 8. While observed HR diagrams (like that plotted for nearby stars in part I)

show the main sequence to have some complex curvature structure, a straight line with

logL ∼ 8 log T does give a rough overall fit, thus providing general support for these simple

scaling arguments.

16.3. Lower mass limit for hydrogen fusion: Brown Dwarf stars

These nuclear burning scalings can also be used to estimate a minimum stellar mass

for Hydrogen fusion. Stars with mass below this minimum are known as Brown Dwarfs. A

key new feature of these stars is that their cores become “electron degenerate”, and so no

longer follow the simple virial scalings derived above for stars in which the pressure is set by

the ideal gas law. Electron degeneracy occurs when the electron number density ne becomes

comparable to cube of the electron De Broglie wavenumber ke ≡ 2π/λe ≡ 1/λe,

ne ≈ k3e =

1

λ3e

, (16.6)

with the electron thermal De Broglie (reduced) wavelength,

λe =~pe

=~√

2mekT, (16.7)

where ~ ≡ h/2π, and the latter equality casts the electron thermal momentum pe in terms

the temperature T and electron mass me. Assuming a constant density ρ = M/(4πR3/3) ≈mpne, we can combine (16.6) and (16.7) with the nuclear temperature (16.4) and the Virial

relation (13.9) to obtain a relation for the stellar mass at which a nuclear burning core should

4Indeed, as already noted, in massive stars the standard, direct proton-proton fusion is augmented by a

process called the CNO cycle, in which CNO elements act as a catalyst for H-fusion. Attaching protons to

such more highly charged CNO nuclei requires a higher core temperature to overcome the stronger electrical

repulsion, and this indeed obtains in such massive stars.

– 16.5 –

become electron degenerate,

Mmin,nuc =

√3/2

4π2

(mp

me

)3/4e3

G3/2m2p

≈ 0.1M . (16.8)

Stars with a mass below this minimum should not be able to ignite H-fusion, because electron

degeneracy prevents their cores from contracting to a small enough size to reach the ∼17 MK

temperature (see eqn. 16.4) required for fusion. In practice, more elaborate computations

indicate such Brown Dwarf stars have a limiting mass MBD ∼< 0.08M, just slightly below

the simple estimate given in (16.8).

Note that, although this minimum mass for H-fusion is limited by electron degeneracy,

the actual value is independent of Planck’s constant h! In effect, the role of h in the tunneling

effect for H-fusion cancels its role in electron degeneracy.

16.4. Upper mass limit for stars: the Eddington Limit

Let’s next consider what sets the upper mass limit for observed stars. This is not linked

to nuclear burning or degeneracy, but stems from the strong L ∼ M3 scaling of luminosity

with mass, which, as noted in §15.1, follows from the hydrostatic support and radiative

diffusion of the stellar envelope.

In addition to its important general role as a carrier of energy, radiation also has an

associated momentum, set by its energy divided by the speed of light c. The trapping of

radiative energy within a star thus inevitably involves a trapping of its associated momentum,

leading to an outward radiative force, or for a given mass, an outward radiative acceleration

grad, that can compete with the star’s gravitational acceleration g. For a local radiative

energy flux F (energy/time/area), the associated momentum flux (force/area, or pressure)

is just F/c. The material acceleration resulting from absorbing this radiation depends on

the effective cross sectional area σ for absorption, divided by the associated material mass

m, as characterized by the opacity κ,

grad =σ

m

F

c=κF

c. (16.9)

For a star of luminosity L, the radiative flux at some radial distance r is just F = L/4πr2.

This gives the radiative acceleration the same inverse-square radial decline as the stellar

gravity, g = GM/r2, meaning that it acts as a kind of “anti-gravity”.

Sir Arthur Eddington first noted that, even for a minimal case in which the opacity just

comes from free-electron scattering, κ = κe = 0.2(1 +X) ≈ 0.34 cm2 g−1 (with the numerical

– 16.6 –

value for standard (solar) Hydrogen mass fraction X ≈ 0.7), there is a limiting luminosity,

now known as the “Eddington luminosity’’, for which the radiative acceleration grad = g

would completely cancel the stellar gravity,

LEdd =4πGMc

κe= 3.8× 104L

M

M. (16.10)

Any star with L > LEdd is said to exceed the “Eddington limit”, since even the radiative

acceleration from just scattering by free electrons would impart a force that exceeds the

stellar gravity, thus implying that the star would no longer be gravitationally bound!

For main sequence stars that follow the L ∼ M3 scaling, setting L = LEdd yields an

estimate for an upper mass limit at which the star would reach this Eddington limit,

Mmax,Edd ≈ 195M , (16.11)

where 195 ≈√

3.8× 104. This agrees quite well with modern empirical estimates for the

most massive observed stars, which are in the range 150-300 M.

Actually, as stars approach this Eddington limit, the radiation pressure alters the hydro-

static structure of the envelope, causing the mass-luminosity relation to weaken toward a lin-

ear scaling, L ∼M , and so allowing in principle for stars with even with mass M > Mmax,Edd

to remain bound. In practice, such stars are subject to “photon bubble” instabilities, much

as occurs whenever a heavy fluid (in this case the stellar gas) is supported by a lighter one

(here the radiation). Very massive stars near this Eddington limit thus tend to be highly

variable, often with episodes of large ejection of mass that effectively keeps the stellar mass

near or below the Mmax,Edd ≈ 195M limit.

Excercise 1:

a. Assume a power-law radius-mass scaling R ∼ Ma for stars on the main-

sequence. Show that there is an associated power-law relation L ∼ T b between

luminosity L and surface temperature T .

b. Give a formula for the power-index b in terms of the index a.

c. Compute the values for b for cases with a = 0.5, 0.7 and 1.

d. What value of a would give the L ∼ T 8 quoted in the text.

– 17.1 –

17. Post-Main-Sequence Evolution

As a star ages, more and more of the Hydrogen in its core becomes consumed by fusion

into Helium. Once this core Hydrogen is exhausted, how does the star react and adjust?

Without the H-fusion to supply its luminosity, one might think that perhaps the star would

simply shrink, cool and dim, and so die out, much as a candle when all its wax is used up.

Instead it turns out that stars at this post-main-sequence stage of life actually start to

expand, at first keeping roughly the same luminosity and so becoming cooler at the surface,

but eventually becoming much brighter giant or supergiant stars, shining with a luminosity

that can be thousands or even tens of thousands that of their core-H-burning main sequence.

Figure 17.1 illustrates the post-MS evolution for the sun (left) and stars with mass up

to 10M (right). Let us focus our initial attention on solar type stars with M . 8M.

Fig. 17.1.— Schematic H-R diagrams to show the post-main-sequence evolution for a solar-

mass star (left), and for stars with M = 1, 5, and 10 M (right).

17.1. Post-MS evolution and death of solar-type stars with M . 8M

17.1.1. Core-Hydrogen burning and evolution to the Red Giant branch

The apparently counterintuitive post-main-sequence adjustment of stars can actually be

understood through the same basic principals used to understand their initial, pre-main-

– 17.2 –

sequence evolution. When the core runs out of Hydrogen fuel, the lack of energy generation

does indeed cause the core itself contract. But the result is to make this core even denser

and hotter. Then, much as the hot coals at the heart of a wood fire help burn the wood fuel

around it much faster, the higher temperature of a contracted stellar core actually makes

the overlying shell of Hydrogen fuel around the core burn even more vigorously!

Now, unlike during the main sequence – when there is an essential regulation or com-

patibility between the luminosity generated in the core and the luminosity that the radiative

envelope is able to transport to the stellar surface –, this shell-burning core is actually over-

luminous relative to the envelope luminosity that is set by the L ∼ M3 scaling law. As

such, instead of emitting this core luminosity as surface radiation, the excess energy acts to

re-inflate the star, in effect doing work against gravity to reverse the Kelvin-Helmholtz con-

traction that occurred during the star’s pre-main-sequence evolution. Initially, the radiative

envelope keeps the luminosity fixed so that, as the star expands, the surface temperature

again declines, with the star thus again evolving horizontally on the H-R diagram, this time

from left to right.

But as the surface temperature approaches the limiting value T ≈ 3500 − 4000 K, the

envelope again becomes more and more convective, which thus now allows this full high-

luminosity of the H-shell-burning core to be transported to the surface. The star’s luminosity

thus increases, with now the temperature staying nearly constant at the cool value for the

Hayashi limit. In the H-R diagram, the star essentially climbs back up the Hayashi track,

eventually reaching the region of the cool, red giants in the upper right of the H-R diagram.

The above describes a general process for all stars, but the specifics depend on the

stellar mass. For masses less than the sun, the main sequence temperature is already quite

close to the cool limit, so evolution can proceed almost directly vertically up the Hayashi

track. For masses much greater than the sun, the luminosity and temperature on the main

sequence are both much higher, and so the horizontal evolutionary phase is more sustained.

And since the luminosity is already very high, these stars become red supergiants without

ever having to reach or climb the Hayashi track.

17.1.2. Helium flash and core-Helium burning on the Horizontal Branch

This Hydrogen-shell burning also has the effect of increasing further the temperature

of the stellar core, and eventually this reaches a level where the fusion of the Helium itself

– 17.3 –

becomes possible, through what’s known as the “triple-α process”5,

3 4He+2 8Be+4 + 4He+2 → 12C+6 . (17.1)

The direct fusion of two 4He+2 nuclei initially make an unstable nucleus of Berylium (8Be+4),

which usually just decays back into Helium. But if the density and temperature are suffi-

ciently high, then during the brief lifetime of the unstable Berylium nucleus, another Helium

can fuse with it to make a very stable Carbon nucleus 12C+6. Since the final step of fusing

4He+2 and 8Be+4 involves overcoming an electrostatic repulsion that is 2×4=8 times higher

than for proton-proton (p-p) fusion of Hydrogen, He-fusion requires a much higher core

temperature, THe ≈ 120 MK.

In stars with more than a few solar masses, this ignition of the Helium in the core occurs

gradually, since the higher core temperature from the addition of He-burning increases the

gas pressure, making the core tend to expand in a way that regulates the burning rate.

In contrast, for the sun and other stars with masses M < 2M, the number density of

electrons ne in the helium core is so high6 that their core becomes electron degenerate. As

discussed in §16.3 for the Brown dwarf stars that define the lower mass limit for H-burning,

electron degeneracy occurs when the mean distance between electrons ∼ n−1/3e becomes

comparable to the DeBroglie wavelength λe = h/pe. The properties of such degeneracy

are discussed further in §17.1.4 on the degenerate white-dwarf end states of solar-ype stars.

But in the present context a key point is that, unlike for the ideal gas law, the pressure

of a degenerate gas becomes independent of temperature! This means that any heating of

the core no longer causes a pressure-driven expansion that normally regulates the nuclear

burning of a core governed by the ideal gas law.

Without then the regulation from such thermal pressure-driven expansion, the ignition

of He burning leads to a Helium flash, in which the entire degenerate core of Helium is fused

into Carbon over a very short timescale. This flash marks the “tip” of the Red Giant Branch

(RGB) in the H-R diagram, but somewhat surprisingly, the sudden addition of energy is

largely absorbed by the overlying stellar envelope. The star thus quickly settles down to a

more quiescent phase of He-burning. Because the He burning causes the core to expand, the

5since Helium nuclei are sometimes referred as “α-particles”

6Recall that on the main sequence the radii of stars is (very) roughly proportional to their mass, R ∼M .

But since density scales as ρ ∼ M/R3, the density of low mass stars tends generally to be higher than in

high-mass stars, roughly scaling as ρ ∼ 1/M2. This overall scaling of average stellar density also applies to

the relative densities of stellar cores, and so helps explain why the cores of low-mass stars tend to become

electron degenerate, while those of higher mass stars do not.

– 17.4 –

shell burning of H actually declines, causing the luminosity to decrease from the tip of the

Red Giant branch, where the He flash occurs, to a somewhat hotter, dimmer region known

as the “Horizontal Branch” in the H-R diagram.

This Horizontal Branch (HB) can be loosely thought of as the He-burning analog of the

H-burning Main Sequence (MS), but a key difference is that it lasts a much shorter time,

typically only 10 to 100 million years, much less than the many billion years for a solar mass

star on the MS. This is partly because the luminosity for HB stars is so much higher than

for a similar mass on the MS, implying a much higher burn rate of fuel. But another factor

is that the energy yield per-unit-mass, ε, for He-fusion to Carbon is about a tenth of that for

H-fusion to Helium, viz. about εHe ≈ 0.06% vs. the εH ≈ 0.7% for H-burning. With lower

energy produced, and higher rate of energy lost in luminosity, the lifetime is accordingly

shorter.

17.1.3. Asymptotic Giant Branch to Planetary Nebula to White Dwarf

Once the core runs out of Helium, He-burning also shifts to an inner shell around the

core, which itself is still surrounded by a outer shell of more vigorous H-burning. This

again tends to increase the core luminosity, but now since the star is cool and thus mostly

convective, this energy is mostly transported to the surface with only a modest further

expansion of the stellar radius. This causes the star to again climb in luminosity along

what’s called the “Asymptotic Giant Branch” (AGB), which parallels the Hayashi track at

just a somewhat hotter surface temperature.

In the sun and stars of somewhat higher mass, up to M . 8M, there can be further

ignition of the Carbon to fuse with Helium to form Oxygen. But further synthesis up the

periodic table requires overcoming the greater electrical repulsion of more highly charge

nuclei. This in turn requires a temperature higher than occurs in the cores lower mass stars,

for which the onset of electron degeneracy prevents contraction to a denser, hotter core.

Further core burning thus ceases, leaving the core as an inert, degenerate ball of C and O,

with final mass on order of 1M, with the remaining mass contained in the surrounding

envelope of mostly Hydrogen.

But such AGB stars tend also to be pulsationally unstable, and because of the very low

surface gravity, such pulsations can over time actually eject the entire stellar envelope. This

forms a circumstellar nebula that is heated and ionized by the very hot remnant core. As

seen in the left panel of figure 17.6, the resulting circular nebular emission glow somewhat

resembles the visible disk of a planet, so these are called “planetary nebulae”, though they

– 17.5 –

really have nothing much to do with actual planets. After a few thousand years, the planetary

nebula dissipates, leaving behind just the degenerate remnant core, a white dwarf star.

17.1.4. White Dwarf stars

The degenerate nature of white dwarf stars endows them with some rather peculiar,

even extreme properties. As noted, they typically consist of roughly a solar mass of C and

O, but have a radius comparable to that of the earth, Re ≈ R/100. This small radius

makes them very dense, with ρwd ≈ 106ρ ≈ 106 g/cm3, i.e. about a million times (!) the

density of water, and so a million times the density of normal main-sequence stars like the

sun. It also gives them very strong surface gravity, with gwd ≈ 104g ≈ 106 m/s2, or about

100,000 times earth’s gravity!

As noted in §16.3 for the Brown dwarf stars that define the lower mass limit for H-

burning, gas becomes electron degenerate when the electron number density ne becomes

so high that the mean distance between electrons becomes comparable to their reduced De

Broglie wavelength,

n−1/3e ≈ λ ≡ ~

pe=

~meve

, (17.2)

where the electron thermal momentum pe equals the product of its mass me and thermal

speed ve, and ~ ≡ h/2π is the reduced Planck constant. The associated electron pressure is

Pe = nevepe = n5/3e

~2

me

=

(ρZ

Amp

)5/3 ~2

me

, (17.3)

where the last equality uses the relation between electron density and mass density, ρ =

neAmp/Z, with Z and Amp the average nuclear charge and atomic mass. For example, for a

Carbon white dwarf, the atomic number Z gives the number of free (ionized) electrons needed

to balance the +Z charge of the Carbon nucleus, while the atomic weight Amp gives the

associated mass from the C atoms. The hydrostatic equilibrium (cf. eqn. 13.1) for pressure

gradient support against gravity then requires for a white dwarf star with mass Mwd and

radius Rwd,PeRwd

≈ ρGMwd

R2wd

. (17.4)

Using the density scaling ρ ∼ Mwd/R3wd, we can combine (17.3) and (17.4) to solve for a

relation between the white-dwarf radius and its mass,

Rwd =1

GM1/3wd

~2

me

(Z

Amp

)5/3

≈ 0.01R

(MMwd

)1/3

, (17.5)

– 17.6 –

where the approximate evaluation uses the fact that for both C and O the ratio Z/A = 1/2.

For a typical mass of order the solar mass, we see that a white dwarf is very compact,

comparable to the radius of the earth, Re ≈ 0.01R. But note that this radius actually

decreases with increasing mass.

17.1.5. Chandasekhar limit for white-dwarf mass: M < 1.4M

This fact that white-dwarf radii decrease with higher mass means that, to provide the

higher pressure to support the stronger gravity, the electron speed ve must strongly increase

with mass. Indeed, at some point this speed approaches the speed of light, ve ≈ c, implying

that the associated electron pressure now takes the scaling (cf. eqn. 17.3),

Pe = necpe = n4/3e ~c =

(ρZ

Amp

)4/3

~c . (17.6)

Applying this in the hydrostatic relation (17.4), we now find that the radius R cancels, and

we instead can solve for a upper limit for a white dwarf’s mass,

Mwd ≤Mch =√

3π

(~cG

)3/2 (Z

Amp

)2

≈ 1.4M , (17.7)

where the subscript refers to “Chandrasekhar”, the astrophysicist who first derived this mass

limit, and the proportionality factor√

3π comes from a detailed calculation beyond the scope

of the discussion here.

As discussed in later sections, when accretion of matter from a binary companion puts a

white dwarf over this limit, it triggers an enormous “white-dwarf supernova” explosion, with

a large, well-defined peak luminosity, L ≈ 1010 L. This provides a very bright standard

candle that can be used to determine distances as far as a Gpc, giving a key way to calibrate

the expansion rate of the universe.

But in the present context, this limit means that sufficiently massive stars with cores

above this mass cannot end their lives as a white dwarf. Instead, they end as violent “core-

collapse supernovae”, leaving behind an even more compact final remnant, either a neutron

star or black hole, as we discuss next.

– 17.7 –

17.2. Post-MS evolution and death of high-mass stars with M > 8M

17.2.1. Multiple shell burning and horizontal loops in H-R diagram

The post-main-sequence evolution of stars with higher initial mass, M > 8M has some

distinct differences from that outlined above for solar and intermediate mass stars. Upon

exhaustion of H-fuel at the end of the main sequence, such stars again expand in radius

because of the over-luminosity of H-shell burning. But the high luminosity and high surface

temperature on the main sequence means that their stellar envelopes remain radiative even

as they expand, never reaching the cool temperatures that force a climb up the Hayashi

track. Instead, their evolution tends to keep near the constant luminosity set by L ∼ M3

scaling for the star’s mass, so evolving horizontally to the right on the H-R diagram.

Since stellar radii scale nearly linearly with mass R ∼M , the mean stellar density ρ ∼M/R3 ∼ M−2 tends to decline with increasing mass. Thus, even after the core contraction

that occurs toward the end of nuclear burning, the core density of high-mass stars never

becomes high enough to become electron degenerate. Moreover the higher mass means a

stronger gravitational confinement that gives higher central temperature and pressure. This

now makes it possible to overcome the increasingly strong electrical repulsion of more highly

charged, higher elements, allowing nucleosynthesis to proceed up the period table all the

way to Iron, which is the most stable nucleus.

However, each such higher levels of nucleosynthesis yields proportionally less and less

energy. This can be seen from the plot in figure 17.2 of the binding energy per nucleon vs.

the number of nucleons in a nucleus. The jump from H to He yields 7.1 MeV, which relative

to a nucleon mass of 931 Mev represents a percentage energy release of about 0.7%, as noted

above. But from He to C the release is just 7.7 – 7.1=0.6 MeV, representing an energy

efficiency of just 0.06%. As the curve flattens out, the fractional energy release become even

less, until for elements beyond Iron, further fusion would require the addition of energy.

For such massive stars, the final stages of post-main sequence evolution are characterized

by an increasingly massive Iron core that can no longer produce any energy by further fusion.

But fusion still occurs in a surrounding series of shells, somewhat like an onion skin, with

higher elements fusing in the innermost, hottest shells, and outer shells fusing lower elements,

extending to an outermost shell of H-burning (see figure 17.3).

– 17.8 –

Fig. 17.2.— Binding energy per nucleon plotted vs. the number of nucleons in a nucleus.

The fusion of light elements moves nuclei to the right, releasing the energy of nuclear burning

in the very hot dense cores of stars, but only up to formation of the most stable nucleus,

for iron (Fe), with atomic number A = 56. In the sudden core collapse of massive-star

supernovae, the copius energy available synthesizes elements even beyond iron. The fission

of such heavy elements leads to lower-mass nuclei toward the left. The energy released is

what powers nuclear fission reactors here on earth.

17.2.2. Core-collapse supernovae

With the build-up of Iron in the core, there is an increasingly strong gravity, but without

the further fusion-generated energy to keep the temperature high, the core pressure becomes

unable to support the mass above. This eventually leads to a catastrophic core collapse,

halted only when the electrons merge with the protons in the Iron nuclei to make the entire

core into a collection of neutrons, with a density so high that they now actually become

neutron degenerate. The “stiffness” of this neutron-degenerate core leads to a “rebound” in

the collapse, with gravitational release from the core contraction now powering an explosion

that blows away the entire outer regions of the star, with the stellar ejecta reaching speeds of

about 10% the speed of light! This ejecta contains Iron and other heavy elements, including

– 17.9 –

Fig. 17.3.— The “onion-skin” layering of the core of a 25-M star just before supernovae

core collapse, illustrating the various stages of nuclear burning in shells around the inert iron

core. The right box shows the decreasing duration for each higher stage of burning.

even those beyond Iron that are fused in less than a second of the explosion by the enormous

energy and temperatures. While elements up to Oxygen can also be synthesized in low-

mass stars, all the heavier elements in our universe, including here on earth, originated in

supernova explosions. For a few weeks, the luminosity of such a supernova can equal or

exceed that of a whole galaxy, up to ∼ 1012L!

Though the dividing line is not exact, it is thought that all stars with initial masses

M > 8M will end their lives with such a core-collapse supernova, instead of following the

track, AGB → PN → White Dwarf, for stars with initial mass M < 8M. Stars with initial

masses 8M < M . 30M are thought to leave behind a neutron star remnant, as discussed

next. But we shall also see that such neutron star remnants have their own upper mass limit

of Mns . 3M, beyond which the gravity becomes so strong that not even the degenerate

pressure from neutrons can prevent a further collapse, this time forming a black hole. This

is thought to be the final core remnant for the most massive stars, those with initial mass

M & 30M.

– 17.10 –

17.2.3. Neutron stars

Neutron stars are even more bizarrely extreme than white dwarfs. With a mass typically

about twice the sun’s, they have a radius comparable to a small city, Rns ≈ 10 km, about

a factor 600 smaller than even a white dwarf, implying a density that is about 108 higher,

and a surface gravity more than 105 times higher.

The radius and mass properties associated with neutron degeneracy can be derived by

a procedure completely analogous to that used in §§17.1.4 and 17.1.5 for white dwarfs, just

substituting now the electron mass with the neutron mass, me → mn ≈ mp, and setting

Z/A = 1. The radius-mass relation thus now becomes (cf. eqn. 17.5),

Rns = 25/3me

mp

Rwd =1

GM1/3ns

~2

m8/3n

≈ 10 km

(MMns

)1/3

. (17.8)

Note again that, as in the case of an electron-degenerate white dwarf, this neutron-star

radius also decreases with increasing mass.

For the same reasons that lead to the upper mass limit for white dwarfs, for sufficiently

high mass the neutrons become relativistic, leading now to an upper mass limit for neutron

stars (cf. eqn. 17.7) that scales as

Mns ≤Mlim = 1.6

(~cG

)3/2 (1

mp

)2

≈ 3M , (17.9)

where again the factor 1.6 comes from detailed calculations not covered here; apart from this

and the factor (∼ A/Z)2 = 4, this is the same form as the Chandrasekhar mass for white

dwarfs in eqn. 17.7. Neutron stars above this mass will again collapse, this time forming a

black hole.

17.2.4. Black Holes

Black holes are objects for which the gravity is so strong that not even light itself can

escape. A proper treatment requires General Relativity, Einstein’s radical theory of gravity

that supplants Newton’s theory of universal gravitation, and extends it to the limit of very

strong gravity. But we can nonetheless use Newton’s theory to derive some basic scalings.

In particular, for a given mass M , a characteristic radius for which the Newtonian escape

speed is equal to speed of light, vesc = c, is just

Rbh =2GM

c2≈ 3 km

M

M, (17.10)

– 17.11 –

which is commonly known as the “Schwarzschild radius”.

Since the speed of light is the highest speed possible, any object within this Schwarzschild

radius of a given mass M can never escape the gravitational binding with that mass. In terms

of Einstein’s General Theory of Relativity, mass acts to bend space and time, much the way

a bowling ball bends the surface of a trampoline. And much as a sufficiently dense, heavy

ball could rip a hole in the trampoline, for objects with mass concentrated within a radius

Rbh, the bending becomes so extreme that it effectively punctures a hole in space-time. Since

not even light can ever escape from this hole, it is completely black, absorbing any light or

matter that falls in, but never emitting any light from the hole itself. This the origin of the

term “black hole”.

Stellar-mass black holes with M & 3M form from the deaths of massive stars. If left

over from a single star, they are hard or even impossible to detect, since by definition they

don’t emit light.

However, in a binary system, the presence of a black hole can be indirectly inferred

by observing the orbital motion (visually or spectroscopically via the Doppler effect) of the

luminous companion star.

Moreover, when that companion star becomes a giant, it can, if it is close enough,

transfer mass onto the black hole. Rather than falling directly into the hole, the conservation

of the angular momentum from the stellar orbit requires that the matter first feed an orbiting

accretion disk. By the Virial theorem, half the gravitational energy goes into kinetic energy

of orbit, but the other half is dissipated to heat the disk, which by the blackbody law then

emits it as radiation.

The luminosity of such black-hole accretion disks can be very large. For a black hole of

mass Mbh accreting mass at a rate Ma to a radius Ra that is near the Schwarzschild radius

Rbh, the luminosity generated is

Ldisk =GMbhMa

2Ra

=Rbh

4Ra

Mac2 ≡ ε Mac

2 . (17.11)

The latter two equalities define the efficiency ε ≡ Rbh/4Ra for converting the rest-mass-energy

of the accreted matter into luminosity. For accretion radii approaching the Schwarzschild

radius, Ra ≈ Rbh, this efficiency can be as high as ε ≈ 0.25, implying 25% of the accreted

matter-energy is converted into radiation. By comparison, for H-fusion of a MS star the over-

all conversion efficiency is about 0.07%, representing the ∼10% core mass that is sufficiently

hot for H-fusion at a specific efficiency, εH = 0.007 = 0.7%.

– 17.12 –

Fig. 17.4.— Artist depiction of mass transfer onto accretion disk around black hole in

Cygnus X-1.

Quick Question 1:

(a) Because of general relativistic effects, it turns out the lowest stable orbit

around a black hole is at radius of 3Rbh. What is the luminosity efficiency for

accreting to this radius?

(b) What is the accretion luminosity, in L, for a mass accretion rate Ma =

10−6M/yr to this radius?

(c) Challenge problem: For a black hole with mass Mbh = 3M, use the

Stefan-Bolzmann law to derive the radiative flux that would balance the local

gravitational heating at this radius r = 3Rbh, and then solve for the local black-

body temperature T (r = 3Rbh). Express this first as a ratio to sun’s surface

temperature T, and then also in Kelvin.

Exercise 1: Use Wien’s law to compute the peak wavelength (in nm) of thermal

emission from the inner region of an accretion disk with temperature T = 107 K.

What is the energy (in eV) of a photon with this wavelength? Now also answer

both questions for T = 1010 K. What parts of the electromagnetic spectrum do

these photon wavlengths/energies correspond to?

In the inner disk, the associated blackbody temperature can reach 107 K or more (see

challenge problem); and at the very inner disk edge, dissipation of the orbital energy can

– 17.13 –

heat material to even more extreme temperatures, up to ∼ 1010 K. The associated radiation

is very high energy, as seen from the Quick-Question calculations. By studying this high-

energy radiation, we can infer the presence and basic properties (mass, even rotation rate)

of black holes in such binary systems, even though we can’t see the black hole itself.

Figure 17.4 shows an artist depiction of the mass transfer accretion in the high-mass

X-ray binary Cygnus X-1, thought to be the clearest example of a stellar-remnant black

hole, estimated in this case to have a mass Mbh > 10M that is well above the Mlim ≈ 3Mupper limit for a neutron star (cf. eqn. 17.9).

Fig. 17.5.— Left: Optical image of the Crab Nebula, showing the remnant from a core-

collapse supernova whose explosion was observed by Chinese astronomers in 1054. Right:

A composite zoomed-in image of the central region Crab Nebula, showing the optical (red)

image superimposed with an X-ray (blue) image made by NASA’s Chandra X-ray observa-

tory. The bright star at the nebular center is the Crab pulsar, a rapidly rotating neutron

star that was left over from the supernova explosion.

17.3. Observations of stellar remnants

It is possible to observe directly all three types of stellar remnants:

1. Planetary Nebula and White Dwarf stars

– 17.14 –

Fig. 17.6.— Left: M57, known at the Ring Nebua, provides a vivid example of a spherically

symmetric Planetary Nebula. The central hot star is the remnant of the stellar core, and

after the nebula dissipates, it will be left as a White Dwarf star. Right: A gallery of planetary

nebulae, showing the remarkable variety of shapes that probably stem from interaction of

the stellar ejecta with a binary companion, or perhaps even with the original star’s planetary

system.

Stars with initial mass M < 8M evolve to an AGB star that ejects the outer stellar

envelope to form a Planetary Nebula (PN) with the hot stellar core with mass below

the Chandrasek mass, Mwd < Mch = 1.4M. Once the nebula dissipates, this leave

behind White Dwarf (WD). White dwarf stars are very hot, but with such a small

radius that their luminosity is very low, placing them on the lower left of the H-R

diagram.

The excitation and ionization of the gas in the surrounding PN makes it shine with

an emission line spectrum, with the wavelength-specific emission of various ion species

giving it range of vivid colors or hues. Figure 17.6 shows that these PN can thus be

visually quite striking, with spherical emission nebula from single stars (left), or very

complex geometric forms (right) for stars in binary systems.

2. Neutron stars and Pulsars

Stars with initial masses in the range 8M < M . 30M end their lives as a core

collapse supernova that leaves behind a neutron star with mass 1.4M < Mns < 3M.

The conservation of angular momentum during the collapse to such a small size (∼10 km) makes them rotate very rapidly, often many times a second! This also generates

– .15 –

a strong magnetic field, and when the polar axis of this field points toward earth, it

leads to a strong pulse of beamed radiation in the radio to optical to even X-rays. This

is observed as a pulsar.

One of the best known examples is the Crab pulsar, which lies at the center of the

Crab Nebula, the remnant from a core-collapse supernova that was observed by Chinese

astronomers in 1054 AD. Figure 17.5 shows images of this Crab nebula in the optical

region (left) and in a composite of images (right) in the optical (red) and X-ray (blue)

wavebands.

3. Black holes and X-ray binary systems

Finally, stars with initial masses M & 30M end their lives with a core collapse

supernova that now leaves behind a black hole with mass Mbh > 3M. As noted, in

single stars, these are difficult or impossible to observe, because they emit no light;

but in binary systems, accretion from the other star can power a bright accretion disk

around the black hole that radiates in high-energy bands like X-rays and even γ-rays.

Figure 17.4 shows an artist depiction of accretion onto a black hole in the high-mass

X-ray binary Cygnus X1.

– C.1 –

C. Radiative Transfer

I(μ,τ=0)

dI=(I-B)dτ/μ

+r

θ

μ=cosθ

+τ

dτ=−κρdr

τ(r=R)=1 R

τ(r=+∞)=0 +∞

dr=μds

+∞

ds dI=(B-I)κρds

Fig. C.1.— Emergent intensity from a semi-infinite, planar atmosphere. Along a direction

s that has projection µ = s · r = cos θ to the local vertical (radial) direction r, the change

in intensity in each differential layer dr depends on thermal emission of radiation by the

local Planck minus the absorption of local intensity I, multiplied by the projected change in

optical depth −dτ/µ = κρds along the path segment ds.

C.1. Absorption and thermal emission in a stellar atmosphere

As noted above (§13.2), the atmospheric transition between interior and empty space

occurs over a quite narrow layer, a few scale heights H in extent, which typically amounts

to about a thousandth of the stellar radius (cf. eqn. 13.5). At any given location on the

spherical stellar surface, the transport of radiation through this atmosphere can be modeled

by treating it as a nearly planar layer, as illustrated in figure C.1.

To quantify this atmospheric transition between random-walk diffusion of the deep inte-

rior to free-streaming away from the stellar surface, we must now solve a differential equation

that accounts for the competition between the reduction in intensity due to absorption vs.

the production of intensity due to the local thermal emission B(τ). As illustrated in figure

C.1, consider a planar atmosphere with an arbitrarily large optical depth (at bottom) seen

from an observer at optical depth zero (at top) who looks along a direction s that has a pro-

– C.2 –

jection7 µ = cos θ to the local vertical (radial) direction r. The change in intensity in each

differential layer dr depends on thermal emission of radiation by the local Planck function

B minus the absorption of local intensity I, multiplied by the projected change in optical

depth −dτ/µ = κρds along the path segment ds. This leads to an “equation of radiative

transfer”,

µdI(µ, τ)

dτ= I(µ, τ)−B(τ) , (C1)

where the radial optical depth integral is now defined from a distant observer at r →∞,

τ(r) ≡∫ ∞r

κρdr′ , (C2)

which thus places the observer at τ(r →∞) = 0.

Fig. C.2.— Visible light picture of the solar disk, showing the center to limb darkening of

the surface brightness.

7This standard notation using µ for direction cosine here should not be confused with the notation in the

previous sections that use µ for molecular weight.

– C.3 –

C.2. The Eddington-Barbier relation for emergent intensity

Eqn. (C1) is a linear, first-order differential equation. As discussed in the exercise below,

by using integrating factors, it can be converted to a formal integral solution for the emergent

intensity seen by an external observer viewing the atmosphere along a projection µ with the

local radius,

I(µ, τ = 0) =

∫ ∞0

B(τ)e−τ/µ dτ/µ ≈ B(τ = µ) . (C3)

The latter approximation here assumes the Planck function is roughly a linear function of

optical depth near the star’s surface, B(τ) ≈ a + bτ . This so-called “Eddington-Barbier

relation” states that when you peer into an opaque radiating gas, the emergent intensity you

perceive is set by the value of the blackbody function at the location of unit optical depth

along that ray. This in turn is set by the temperature at that location, providing a more

rigorous definition for what we’ve referred to up to now as surface brightness and surface

temperature.

An example of this E-B relation comes from the observed “limb darkening” of the solar

disk, as illustrated by the visible light picture of the sun in fig. C.2. Because the line of

sight looking at the center is more directly radial to the sun’s local surface, one can see into

a deeper, hotter layer than from the more oblique angle when viewing toward the edge or

“limb” of the solar disk. This makes the disk appear brightest at the center, and darker

as the view moves toward the solar limb8. The observed variation from center to limb thus

provides a diagnostic of the temperature gradient in the sun’s surface layers.

Since stars are too far away to resolve their angular size, we can’t observe their emergent

intensity I(µ, 0), but we can observe the flux F (r) = L/4πr2 associated with the total lumi-

nosity L = 4πR2σsbT4eff . The emergent surface flux F∗ = L/4πR2 is obtained by integrating

8Of course, the brightness of the sun means we need special filters to see this effect. One should never

look at the sun with the naked eye.

– C.4 –

µI(µ, 0) over the 2π solid angle for the hemisphere open to empty space, giving

F∗ ≡ 2π

∫ 1

0

µI(0, µ)dµ

≈ 2π

∫ 1

0

µB(τ = µ)dµ

≈ 2π

∫ 1

0

µ(a+ bµ)dµ

= πB(τ = 2/3)

= σsbT4(τ = 2/3) , (C4)

where the third equality assumes the Planck function near the surface can be approximated

at a linear function of optical depth, B(τ) ≈ a+ bτ .

Comparison of the final form of (C4) with the simple discussion of surface flux in part

I shows that we can identify what we’ve been calling the stellar “surface” as the layer where

the optical depth τ(R) ≡ 2/3, with the “surface temperature” likewise just the temperature

at this layer.

Stars are not really black-bodies, but it is convenient to define a star’s “effective tem-

perature” Teff as the blackbody temperature that would give the star’s inferred surface flux

F∗ = L/4πR2. From (C4), we see that we can associate this effective temperature with the

surface temperature at optical depth 2/3, Teff = T (τ = 2/3).

C.3. Questions and Exercises

Exercise 1: Derive the integral solution (C3) from the differential equation (C1),

assuming a semi-infinite atmosphere that extends to large depths τ →∞. Hint:

First multiply eqn. (C1) by an integrating factor e−τ/µ, and use this to write the

change in intensity in terms of a full differential. Then carry out the integral

from the observer at τ = 0 to some finite depth τ where the intensity is taken to

have a given value I(τ, µ). Finally take the limit τ →∞ to obtain (C3).

Exercise 2: If thermal emission from the Planck function is a linear function of

radial optical depth B(τ) = a+ bτ , explicitly do the integration in (C3) to derive

the Eddington-Barbier relation for emergent intensity I(µ, 0) = B(τ = µ).

– D.0 –

D. Atomic origins of opacity

For solid objects in our everyday world, the interaction with light depends on the object’s

physical projected area, which is the source of the above concept of a “cross section”. But as

noted in §12.3, for interstellar dust with sizes become comparable to the wavelength of light,

the effective cross section can depend on this wavelength, and so differ from the projected

geometric area.

For atoms, ions and electrons that make up a gaseous object like a star, the effective

cross sections for interaction with light can be even more sensitive to the details. But

generally because light is an Electro-Magnetic (EM) wave, at the atomic level its fundamental

interaction with matter occurs through the variable acceleration of charged particles by the

varying electric field in the wave. As the lightest common charged particle, electrons are

most easily accelerated, and thus are generally key in setting the interaction cross section.

The simplest example is that of an isolated free electron, so let’s begin by examining its

interaction cross section and opacity.

D.1. Thomson cross-section and opacity for free electron scattering

As illustrated in the top left panel of figure D.1, when a passing EM wave causes a free

electron to oscillate, it generates a wiggle in the electron’s own electric field, which then

propagates away – at the speed of light – as a new EM wave in a new direction. Because an

isolated electron has no way to store both the energy and momentum of the incoming light,

it cannot by itself absorb the photon, and so instead simply scatters, or redirects it. The

overall process is called “Thomson scattering”.

For such free electrons, the associated Thomson cross section can actually be accurately

computed using the classical theory of electromagnetism. Intuitively, the scaling can be

roughly understood in terms of the so-called “classical electron radius” re = e2/mec2, which

is just the radius at which the electron’s electrostatic self-energy e2/re equals the electron’s

rest-mass energy mec2. In these terms, the Thomson cross section for free-electron scattering

is just a factor9 8/3 times greater than the projected area of a sphere with the classical

electron radius,

σTh =8

3πr2

e =8

3

πe4

m2ec

4= 0.66× 10−24 cm2 . (D1)

9This factor 8/3 comes from detailed classical calculations, and is not easy to understand in simple

intuitive terms.

– D.1 –

e-

e-

e-

+

e-

+

e-

+

e-e-

+

e-

+

e-+

e-

+

Scattering

Thomson/free-free

Absorption Emission

bound-bound

bound-free/free-bound

Fig. D.1.— Illustration of the free-electron and bound-electron processes that lead to

scattering, absorption, and emission of photons.

For stellar material to have an overall neutrality in electric charge, even free electrons

must still be associated with corresponding positively charged ions, which have much greater

mass. Defining then a mean mass per free electron µe, we can also define an electron

scattering opacity κe ≡ σTh/µe. Ionized hydrogen gives one proton mass mp per electron,

but for fully ionized Helium (and indeed for most all heavier ions), there are two nucleon

masses (one proton and one neutron, mp +mn ≈ 2mp) for each electron. For ionized stellar

material with hydrogen mass fraction X, we thus have µe = 2mp/(1 +X), which then gives

for the opacity,

κe ≡σThµe

= 0.2 (1 +X) cm2/g = 0.34 cm2/g , (D2)

where the last equality assumes a “standard” solar Hydrogen mass fraction X = 0.72.

– D.2 –

D.2. Atomic absorption and emission: free-free, bound-bound, bound-free

When electrons are bound to atoms or ions, or even just nearby ions, then the combi-

nation of the electron and atom/ion can lead to true absorption of a photon of light. As

shown in the center top row of figure D.1, for free electrons near ions, the shift in the electron

trajectory as it passes an ion can now absorb a photon’s energy, a process called free-free ab-

sorption. The right top panel shows that the inverse process can actually produce a photon,

and so is called free-free emission.

The second row illustrates bound-bound processes, involving up/down jumps of electrons

between two bound energy levels of atom, with associated absorption/emission of photon

(middle and right panel in second row), or indeed, a scattering if the absorption is quickly

followed by a reemission of a photon with the same energy, but in a different direction (left

panel, second row).

These bound-bound processes only work with photons with just the right energy to

match the difference in energy levels, and so lead to the spectral line absorption or emission

discussed earlier. But for those “just right” photons, the interaction cross section (leading to

the opacity) can be much much higher than for Thomson scattering or free-free absorption,

because in effect it is a kind of “resonance” interaction. An everyday analogy is blowing into

a whistle vs. just into open air. In open air, you get a weak white noise sound, made up of a

range of sound frequencies/wavelengths. With a whistle, the sound is loud and has a distinct

pitch, representing a resonance oscillation at some well-defined frequency/wavelength.

The third row illustrates the bound-free processes associated with a photon absorption

that causes an atom or ion to become (further) ionized by kicking off its electron. As with

electron scattering or even free-free absorption, it is a continuum (vs. line) process, though

it does now require that the photons have a energy equal to or greater than the ionization

energy for that atom or ion. Its interaction cross-section can be significantly higher than

electron scattering or free-free absorption, but is generally not as strong as for bound-bound

processes that lead to lines.

The cross sections, and corresponding opacities, associated with these electron+ion/atom

processes are much more complicated than for free electrons, and so are difficult to cast in

the kind of simple formula given in eqn. D2 for Thomson electron scattering opacity. But

often bound-free and free-free opacities are taken to follow a so-called “Kramer’s opacity”

for which

κkr ∼ ρT−7/2 ∼ (Pgas/Prad)T−1/2 . (D3)

As discussed further below, often in stellar interiors the ratio of gas to radiation pressure

is nearly constant, so that opacity decreases only weakly (as 1/√T ) with the increasing

– D.3 –

temperature of the interior.

Some explicit opacity scaling formulae are given by:

www.astro.psu.edu/users/rbc/a534/lec6.pdf

A simple rough rule of thumb is that, outside of ionization zones where bound-free ab-

sorption can substantially enhance the overall opacity, stellar interiors typically have opacities

that are some modest factor few times the simple electron scattering opacity in (D2), i.e.,

with a characteristic CGS value of order unity, κ ≈ 1 cm2/g.

D.3. Questions and Exercises

Quick Question 1:

a. Seen standing up, what is the cross section (in cm2) of a person with height

1.8 m and width 0.5 m?

b. If this person has a mass of 60 kg, what is his/her “opacity” κ = σ/m, in

cm2/g?

c. How does this compare with the typical opacity of stellar material?